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We present a general framework for obtaining robust bounds on the nature of dark matter using cosmological $N$-body simulations and Lyman-alpha forest data. We construct an emulator of hydrodynamical simulations, which is a flexible, accurate and computationally-efficient model for predicting the response of the Lyman-alpha forest flux power spectrum to different dark matter models, the state of the intergalactic medium (IGM) and the primordial power spectrum. The emulator combines a flexible parameterization for the small-scale suppression in the matter power spectrum arising in "non-cold" dark matter models, with an improved IGM model. We then demonstrate how to optimize the emulator for the case of ultra-light axion dark matter, presenting tests of convergence. We also carry out cross-validation tests of the accuracy of flux power spectrum prediction. This framework can be optimized for the analysis of many other dark matter candidates, e.g., warm or interacting dark matter. Our work demonstrates that a combination of an optimized emulator and cosmological "effective theories," where many models are described by a single set of equations, is a powerful approach for robust and computationally-efficient inference from the cosmic large-scale structure.